About transmission eigenstates

[xhsh]

From here: http://www.quantumwise.com/documents/manuals/latest/ReferenceManual/index.html/ref.transmissioneigenstate.html

we are told that the transmission eigenstates are linear combinations of Bloch states of the electrodes.  However, I think this definition about transmission eigenstates there is wrong or at least misleading or confusing. Obviously, the Bloch states only belong to one electrode. With the Bloch state of only the left electrode, we can not get the spatial distribution of the transmission eigenstate in the central region and the right electrode since these Bloch states are not defined there.  Thus, I think, the transmission eigenstates are the linear combinations of the scattering states which are distributed in the whole space and span a Hilbert space. The scattering states  are defined by equation (4.1) in Jeremy Talor's PhD thesis.

Please correct me if I am wrong and please give a more detailed description or introduction about it. Many thanks.

[Petr Khomyakov]

The  manual on TransmissionEigenstate does not say that "the transmission eigenstates are linear combinations of Bloch states of the electrodes".

In the manual section you refer to, it is said that "t_{nk} is the transmission amplitude from Bloch state \psi_n in the left electrode to Bloch state \psi_k in the right electrode.
 ...
The transmission eigenstates are obtained by propagating the linear combination of the Bloch states".

It means that the transmission eigenstates are the linear combination of the Bloch states in the semi-infinite electrodes, not the central region. The corresponding electron wave function given by this linear combination of the Bloch states is then effectively matched to the electron wave function in the central region by propagation through the central region.

A good textbook example is the electron scattering on a square barrier potential, where the wave function in the electrodes is a linear combination of an incoming and reflected Bloch state of the electron, which are just two plane waves in this case indeed. The under barrier wave function is a simple decaying function that has to be matched to the wave function in the electrodes.

You may solve the same problem without explicit wave-function matching, e.g., by propagation of the corresponding states of the electrodes through the central region. And this is what is described in Jeremy's thesis as well as in most dissertations on quantum transport.

 

[xhsh]

Many thanks for your reply. I still do not understand very well. Several questions:

1. Is the T matrix just the matrix Gamma_LG^RGamma_RG^A?

2. By "propagating the linear combination of the Bloch states" , do you mean  we take the linear combinations of the Bloch states of one lead as  an incoming wave and we calculate the reflected wave in the left lead and the transmitted wave in the right lead and the wave in the central region, by the same procedure as calculating the scattering states in Jeremy's thesis?

3.  How are the transmission eigenstates calculated in detail and do you have any reference for it, please?

4. Do the Bloch states here include all the propagating modes and evanescent modes which are obtained by solving a complex band structure problem? That is, are the incoming waves the linear combinations of all these modes? Do the transmission eigenstates in the left lead include the left moving modes?

[Petr Khomyakov]

There is no really short answer to your questions. You may however find them as well as answer to other potential questions on quantum transport if you read, for example, the following two papers: http://journals.aps.org/prb/abstract/10.1103/PhysRevB.72.035450 and http://journals.aps.org/prb/abstract/10.1103/PhysRevB.70.195402 . 
There are many other publications on the topic; these two are just my favorite.

[xhsh]

Many thanks  to your suggestions.